Ampere–Oersted field splitting of the nonlinear spin-torque vortex oscillator dynamics

We investigate the impact of the DC current-induced Ampère–Oersted field on the dynamics of a vortex based spin-torque nano-oscillator. In this study we compare micromagnetic simulations performed using mumax\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^3$$\end{document}3 and our analytical model based on the Thiele equation approach. The latter is improved by adding two important corrections to the Thiele equation approach. The first is related to the magneto-static contribution and depends on the aspect ratio of the magnetic dot. The second is a full analytical description of the Ampère–Oersted field contribution. The model describes quantitatively the simulation results in the resonant regime as well as the impact of the Ampère–Oersted field. Depending on the relative orientation between the vortex in-plane curling magnetisation (chirality) and the Ampère–Oersted field a strong splitting phenomenon appears in the fundamental properties (frequency and vortex core position) of the nano-oscillator. Thus, we show that the Ampère–Oersted field should not be neglected as it has a high impact on the spin-torque vortex oscillator dynamics.


Ampère-Oersted field contribution to the Thiele equation
Ampère's law (SI): where k m is the magnetic force constant (µ 0 /(4π)) and µ 0 is the magnetic constant [4π · 10 −7 T/(A/m)]. Let's take the case of an infinite wire of radius R with a uniform current I flowing through it. I c represents the current flowing inside the path c of integration.
Inside the wire (r R): I c = Iπr 2 /(πR 2 ) = Ir 2 /R 2 = πJr 2 , where J = I/(πR 2 ) is the current density. Outside the wire (r > R): The Oersted field vector obtained is: B(r) = B(r) cos θ + σ π 2 , sin θ + σ π 2 , 0 = B(r)b(r), where r = r(cos θ , sin θ , 0) or r = (r, θ ), σ = sign(J) = ±1 and b(r) is the Oersted field unit vector. Inside a magnetic dot of radius R and thickness h, the potential energy due to the current induced Oersted field and a shifted magnetic vortex state of magnetisation distribution M(r, X) with X = (ρ, ϕ) being the vortex core position is given by: where V corresponds to the magnetic dot volume.

One obtains explicitly:
φ TVA 2 (r, X) = tan −1 r sin θ − ρ sin ϕ r cos θ − ρ cos ϕ + tan −1 ρr sin θ − R 2 sin ϕ ρr cos θ − R 2 cos ϕ − ϕ +C π 2 The cylindrical symmetry of the energy evaluation of an off-centred vortex with respect to the Oersted field makes it independent from ϕ, so we choose to take ϕ = 0: Finally, using reduced variables η = r/R and s = ρ/R we obtain: The previous transformation is due to the fact that there is no z-dependence.
As η = r/R, dr = Rdη: Finally, the potential energy writes (σ |J| = J): There are two possibilities to evaluate Eq. (5) to obtain the s = ρ/R dependence of W Oe . The first is to numerically integrate it and then to fit the result to a power law in s. The second possibility is to compute the Taylor expansion (TE) of the integrand of Eq. (5) and then solve it analytically. Both techniques are considered and compared hereafter. Figure S1 shows the numerical resolution of the integral in Eq. (5) in blue. The 6 th order power law fit and the 10 th order Taylor expansion are represented in blue and red dashed lines, respectively. | | Figure S1. Evolution of the double integral in Eq. (5) vs. reduced vortex core position s. The thick light blue line corresponds to the numerical integration, the dashed blue line is the 6 th order fit over the numerical data and the dashed red line is the result after analytical integration of the 10 th order Taylor expansion.
It should be noticed that 2/3π has been subtracted from the overall value of the integral shown in Fig. S1 in such a way to start at zero and avoid the evaluation of this parameter during the fit. The fitting coefficients are given as follows: W Oe fit (s) = −QJCM s hR 3 2 3 π − 0.827s 2 + 0.180s 4 + 0.119s 6 .
The 10 th order Taylor expansion gives: Figure S2 shows the error level between the numerical integration data and both, the fit [Eq. (6)] and the 10 th order Taylor expansion [Eq. (7)].
An important consequence of the error level comparison is the fact that the 10 th order TE is much more accurate than the fit for s 0.78, but for s > 0.78 the fit maintains a lower error level.
As mentioned in the main text of this manuscript, the vortex is unstable for s > 0.8, so the Taylor expansion is a better choice for modelling the Ampère-Oersted field contribution to the Thiele equation.
The current induced Oersted field acts as a restoring force and one can thus consider: where X (= ρ) is the vortex core orbital radius and k Oe is the restoring force constant (also called vortex stiffness parameter). The corresponding restoring force magnitude writes: The energy W Oe in Eqs. (6) and (7) is already depending on s, so we can rewrite Eq. (9) as follows to consider the force in terms of the reduced vortex core position s: Eqs. (6) and (7) then give the following results for the respective restoring force constants: In addition, one can redefine the spring-like restoring force constant associated to the Ampère-Oersted field as k Oe (s) = κ Oe (s)JC for the sake of clarity. This transformation, used in the main text of this manuscript, allows to highlight the impact of the current density and the chirality on the oscillator dynamics.
Finally, the oscillation frequency contribution of the Oersted field is given by ω Oe = k Oe /G.

5/7
The magneto-static contribution W ms (ξ , s) to the Thiele equation has been calculated by Gaididei et al. 1 under the "Two Vortex Ansatz" and gives the following equation: with  Table S1.
There is no analytical solution for Eq. (10) and a deterministic numerical integration is not feasible. So, we performed a Monte Carlo (MC) integration (non-deterministic) with guaranteed absolute precision of 10 Even if the practical magnetic dot geometries give rise to very small ξ values, the latter are not zero. For instance in this manuscript, R = 100 nm and h = 10 nm, so ξ = h/(2R) = 0.05. The Monte Carlo integration for Θ(ξ = 0.05, s) gives rise to: k ms ξ =0.05 (s) = 8M 2 s h 2 R 1.5941 1 + 0.175s 2 + 0.065s 4 − 0.054s 6 .
(12) 1 The library used for the MC integration is GAIL version 2.0 (Guaranteed Automatic Integration Library) developed by the Illinois Institute of Technology.

6/7
The mean relative overestimation of Eq. (11) compared to Eq. (12) is about 10.3%. It goes from 9.3% at s = 0 to 13.2% at s = 0.8. As the oscillation frequency contribution of the magneto-static energy is given by ω ms = k ms /G, the vortex core gyrotropic frequencies computed using ξ = 0 instead of ξ = 0.05 in this case give rise to a mean error of about 10.3%.  Table S1. Values of the coefficients of the k ms term after the fit and derivation according to the power law model given in the main text.